Practicing Success
In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor a car. 2000 families own both a car and a phone. Consider the following statements in this regard: |
1 and 2 1 and 3 2 and 3 1, 2 and 3 |
2 and 3 |
The correct answer is Option (3) → 2 and 3 Let there be x families in the town. Let P and C denote the set of families using phone and car respectively. Then, $n(P) =\frac{25x}{100}=\frac{x}{4}, n(C)=\frac{15x}{100}=\frac{3x}{20}$ and, $n(\overline P∩\overline C)=\frac{65x}{100}=\frac{13x}{20}$ Also, $n(P∩C) = 2000$. Now, $n(\overline P∩\overline C)=\frac{13x}{20}$ $⇒n(\overline{P∪C})=\frac{13x}{20}$ $⇒n(U)-n(P∩C)=\frac{13x}{20}$ $⇒x-\{n(P)+n(C)-n(P∩C)\}=\frac{13x}{20}$ $⇒x-\left(\frac{x}{4}+\frac{3x}{20}-2000\right)=\frac{13x}{20}⇒\frac{x}{20}=2000⇒x=40000$ Thus, statement 3 is true. 10% of the total families in the town is 4000 and it is given that 2000 families own both a car and a phone. So, statement 1 is not true. Now, $n(P∪C)=n (P) + n (C) -n (P∩C)$ $⇒n(P∪C)=\frac{x}{4}+\frac{3x}{20}- 2000$ $⇒n(P∪C)=10000+ 6000 - 2000 = 14000$ $⇒n(P∪C)$ = 35% of 40000 Thus, statement 2 is correct. |